Then it might not be so odd. Just to be clear, Chromium is just an open source, constantly
changing in-development code-testing browser project used for the development of Chrome browser.
Chromium itself is not a browser, and Google/Chromium-devs do not expect Chromium to be run as a
stable, final use browser. What many Linux distros do (including Ubuntu), and compile and release
Chromium executables, is beyond what Chromium is designed for and is the responsibility of each distro.
You can find more contact info for each distro's Chromium from:
http://code.google.com/p/chromium/wiki/LinuxChromiumPackages
As @ashejole pointed out in Comment 2, you're asked to try from official Chrome browser. Are you
sure the coding for navigator.geolocation.getCurrentPosition is available in open-source Chromium
or that the Ubuntu distro didn't mess it up (since it works with official Chrome)?

For anybody else coming across with this, the fix should be coming out when Chromium 26 is released to the repos. In the meantime, I've figured out a workaround involving setting up your own Google API keys if you really need geolocation working before the new release is out. I've posted some notes in launchpad here: https://bugs.launchpad.net/ubuntu/+source/chromium-browser/+bug/1174414

I found the culprit:
http://www.chromium.org/developers/how-tos/api-keys
Apparently since chromium 23, some features were slowly moved to require Google API keys in order to function. In a precompiled version of chrome or chromium nightly or canary, everything seems to work, because these builds use default keys. But if we build chromium by ourselves, it won't have any keys and functions like Google Maps Geolocation API, Sync API, etc. won't function properly. You may also get this message when starting Chromium:
Google API keys are missing. Some functionality of Chromium will be disabled.
A simple example to reproduce the missing keys is to get an official Google Chrome build, Geolocation API should work. Then open Command Prompt and run these commands to define environment variables forcing the official Chrome to use custom keys (in this example invalid keys to break the geolocation service):
set GOOGLE_API_KEY=aaaaaaaaaa
set GOOGLE_DEFAULT_CLIENT_ID=1111111.apps.googleusercontent.com
set GOOGLE_DEFAULT_CLIENT_SECRET=ccccc
start chrome.exe
Then any attempt to use the Geolocation API will fail:
Network location provider at 'https://www.googleapis.com/' : Returned error code 403.
I was able to get most of the functions to work by following the steps on the first link and once I acquired my own keys I defined the environment variables above to test them and they worked. Then I integrated my keys inside my chromium build, by editing this file:
src/build/common.gypi
'google_api_key%': 'ENTER API KEY HERE',
'google_default_client_id%': 'ENTER CLIENT ID HERE.apps.googleusercontent.com',
'google_default_client_secret%': 'ENTER CLIENT SECRET HERE',
The build was successful and my keys work. I no longer need to define the environment variables. The problem is that Geolocation is still returning the same error. And as I read, the problem is that this API requires billing to be enabled, and I haven't done so, hence it still doesn't work:
Google Maps Geolocation API
(requires enabling billing but is free to use; you can skip this one, in which case geolocation features of Chrome will not work)
As I read in src/build/common.gypi
# You can set the variable 'use_official_google_api_keys' to 1
# to use the Google-internal file containing official API keys
# for Google Chrome even in a developer build. Setting this
# variable explicitly to 1 will cause your build to fail if the
# internal file is missing.
I have to test this and hopefully I will be able to integrate the default keys and use geolocation.

Setting 'use_official_google_api_keys' to 1 only works if you have the Google-internal bits needed to make an official Google Chrome build. Nobody outside of Google has access to those, so the last thing you suggested to test will not work for you.
AFAIK, the only way to get geolocation to work in your personal development Chromium build would be to enable billing as mentioned for the Google Maps Geolocation API.